LESSON
19.4
Name
Perpendicular Lines
Class
19.4 Perpendicular Lines
Essential Question: What are the key ideas about perpendicular bisectors of a segment?
Common Core Math Standards
The student is expected to:
1
G-CO.9
Prove theorems about lines and angles. Also G-CO.12, MP.3.1
Explore
MP.5 Using Tools
A
Explain to a partner why a pair of lines is or is not perpendicular.
ENGAGE
Place the point of the compass
at point A. Using a compass
setting that is _
greater than half
the length of AB, draw an arc.
B
Without adjusting the compass,
place the point of the compass at
point B and draw an arc intersecting
the first arc in two places. Label the
points of intersection C and D.
C
B
Use a _
straightedge to
draw CD, which is the
perpendicular
bisector
_
of AB.
C
Essential Question: What are the key
ideas about perpendicular bisectors of a
segment?
View the Engage section online. Discuss the photo.
Ask students how they would balance the competing
factors that a power company might face in deciding
where to build a wind farm. Then preview the Lesson
Performance Task.
A
¯.
In Steps A–C, construct the perpendicular bisector of AB
Language Objective
A
B
C
A
B
A
B
D
D
In Steps D–E, construct a line perpendicular to a line ℓ that passes through
some point P that is not on ℓ.
© Houghton Mifflin Harcourt Publishing Company
PREVIEW: LESSON
PERFORMANCE TASK
Resource
Locker
Constructing Perpendicular Bisectors
and Perpendicular Lines
You can construct geometric figures without using measurement tools like a ruler or a
protractor. By using geometric relationships and a compass and a straightedge, you can
construct geometric figures with greater precision than figures drawn with standard
measurement tools.
Mathematical Practices
Possible answer: If you know that a line is the
perpendicular bisector of a segment, then any point
on the line is equidistant from the endpoints of the
segment. If you know that a point on a line is
equidistant from the endpoints of a segment, then
the line must be the perpendicular bisector of the
segment.
Date
D
E
Place the point of the compass at P.
Draw an arc that intersects line ℓ at
two points, A and B.
P
ℓ
Use the methods in Steps A–C to
construct
the perpendicular bisector
_
of AB.
P
ℓ
A
P
ℓ
B
A
B
_
Because it is the perpendicular bisector of AB, then the constructed
line through P is perpendicular to line ℓ.
Module 19
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Lesson 4
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Lesson 4
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Module 19
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IN1_MNL
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Lesson 19.4
4/19/14
12:41 PM
9/23/14 9:04 AM
Reflect
1.
In Step A of the first construction,
_ why do you open the compass to a setting that is
greater than half the length of AB?
This ensures that the two arcs will intersect at two points.
2.
What If? Suppose Q is a point on line ℓ. Is the construction of a line perpendicular
to ℓ through Q any different than constructing a perpendicular line through a point P
not on the line, as in Steps D and E?
Constructing the points A and B on line ℓ is different in the two constructions. For a
EXPLORE
Constructing Perpendicular Bisectors
and Perpendicular Lines
INTEGRATE TECHNOLOGY
point Q on line ℓ, you place the compass point at Q and draw arcs on either side of Q. The
Students have the option of exploring the
perpendicular bisector activity either in the book
or online.
intersection points will be points A and B. Then, you can follow the same methods as in
Steps A–C in the Explore.
Proving the Perpendicular Bisector Theorem
Using Reflections
Explain 1
QUESTIONING STRATEGIES
You can use reflections and their properties to prove a theorem about perpendicular bisectors.
These theorems will be useful in proofs later on.
Why do you have to use a compass setting
greater than half the length of the segment
when you construct the perpendicular bisector of the
segment? This ensures that the two arcs will
intersect.
Perpendicular Bisector Theorem
If a point is on the perpendicular bisector of a segment, then it is equidistant from
the endpoints of the segment.
Prove the Perpendicular Bisector Theorem.
_
Given: P is on the perpendicular bisector m of AB.
Example 1
Prove: PA = PB
m
line m
. Then the reflection
of point P across line m is also
P
because point P lies
on
A
B
line m
, which is the line of reflection.
Also, the reflection of
point A
across line m is B by the definition
of reflection .
Therefore, PA = PB because reflection preserves distance.
Reflect
3.
Discussion What conclusion can you make about △KLJ in the diagram using
the Perpendicular Bisector Theorem?
K
M
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Critical Thinking
MP.3 Have students think about the steps used with
© Houghton Mifflin Harcourt Publishing Company
P
Consider the reflection across
each construction. Ask them to reflect on how they
remember how to do each. In small groups, have
students discuss strategies for remembering the steps.
Then share each group’s best strategies with the class.
EXPLAIN 1
L
JK = JL because point J lies on the perpendicular
_
bisector of KL.
Proving the Perpendicular Bisector
Theorem Using Reflections
J
Module 19
966
Lesson 4
QUESTIONING STRATEGIES
PROFESSIONAL DEVELOPMENT
IN1_MNLESE389762_U8M19L4 966
Integrate Mathematical Practices
This lesson provides an opportunity to address Mathematical Practice MP.5,
which calls for students to “use appropriate tools.” They begin the lesson by
constructing the perpendicular bisector of a segment; they may also use paperfolding or reflective devices to construct the perpendicular bisector. Students also
construct the perpendicular to a line through a point not on the line. For each
construction, they must be able to use the tools in a variety of ways in order to
obtain accurate results and to understand the underlying mathematical
relationships.
4/19/14 12:40 PM
Suppose you construct the perpendicular
bisector of a segment and then choose any
point on the perpendicular bisector. If you measure
the distance from the point to each endpoint of the
segment, what do you expect to find? The distances
from the point to each endpoint of the segment
are equal.
Perpendicular Lines 966
Your Turn
_
¯.
Use the diagram shown. BD is the perpendicular bisector of AC
AVOID COMMON ERRORS
D
When finding the distance from a point on one side
of a perpendicular bisector to its reflected point,
students may give the distance to the bisector as the
solution. Remind students to re-read the question to
verify that they have answered it by using given
information, such as congruence markings.
A
Suppose ED = 16 cm and DA = 20 cm. Find DC.
Because ¯
BD is the perpendicular bisector of ¯
AC, then DA = DC and DC = 20 cm.
4.
E
C
5. Suppose EC = 15 cm and BA = 25 cm. Find BC.
Because ¯
BD is the perpendicular bisector of ¯
AC, then BA = BC and BC = 25 cm.
B
Proving the Converse of the Perpendicular
Bisector Theorem
Explain 2
The converse of the Perpendicular Bisector Theorem is also true. In order to prove the converse,
you will use an indirect proof and the Pythagorean Theorem.
In an indirect proof, you assume that the statement you are trying to prove is false. Then you use
logic to lead to a contradiction of given information, a definition, a postulate, or a previously proven
theorem. You can then conclude that the assumption was false and the original statement is true.
EXPLAIN 2
Proving the Converse of the
Perpendicular Bisector Theorem
c
a
a2 + b2 = c2
Recall that the Pythagorean Theorem
states that for a right triangle with legs
of length a and b and a hypotenuse
of length c, a 2 + b 2 = c 2.
b
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Communication
MP.3 Make sure students understand that the proof
1. Identify the conjecture to be proven.
2. Assume the opposite of the conclusion is true.
3. Use direct reasoning to show the assumption leads
to a contradiction.
4. Conclude that since the assumption is false, the
original conjecture must be true.
Example 2
_
Prove: P is on the perpendicular bisector m of AB.
Near the end of an indirect proof, a step
contradicts a known true statement. What
does this mean in terms of the proof? The original
assumption is false, so what you are trying to prove
must be true.
967
Lesson 19.4
m P
Step A: Assume what you are trying to prove is false.
A
¯
AB
Assume that P is not on the perpendicular bisector m of
.
Then, when you draw a perpendicular line from P to the line containing A and B,
_
_
it intersects AB at point Q, which is not the midpoint of AB.
Q
B
Step B: Complete the following to show that this assumption leads to a contradiction.
_
△BQP
PQ forms two right triangles, △AQP and
.
So, AQ 2 + QP 2 = PA 2 and BQ 2 + QP 2 = PB 2 by the Pythagorean Theorem.
Subtract these equations:
AQ 2 + QP 2 = PA 2
B
Q 2 + QP 2 = PB 2
____
2
AQ 2 - BQ 2 = PA - PB 2
However, PA 2 - PB 2 = 0 because
QUESTIONING STRATEGIES
How can you tell that an indirect proof is
used? In the first step, you assume that what
you are trying to prove is false.
Prove the Converse of the Perpendicular Bisector Theorem
Given: PA = PB
© Houghton Mifflin Harcourt Publishing Company
of the Converse of the Perpendicular Bisector
Theorem is based on the method of indirect proof. To
write an indirect proof:
Converse of the Perpendicular Bisector Theorem
If a point is equidistant from the endpoints of a segment, then it lies on the
perpendicular bisector of the segment.
PA = PB
.
2
2
Therefore, AQ 2 - BQ 2 = 0. This means
_that AQ = BQ and AQ = BQ. This contradicts
the fact that Q is not the midpoint of AB. Thus, the initial assumption must be incorrect,
_
and P must lie on the perpendicular bisector of AB.
Module 19
967
Lesson 4
COLLABORATIVE LEARNING
IN1_MNLESE389762_U8M19L4 967
Whole Class Activity
Have groups of students create posters to describe how to do an indirect proof.
Then demonstrate the method for the Converse of the Perpendicular Bisector
Theorem. Ask them to display their results as a graphic organizer that shows the
steps for the proof.
9/23/14 9:06 AM
Reflect
6.
COMMUNICATING MATH
In the proof, once you know AQ 2 = BQ 2, why can you conclude that AQ = BQ?
Take the square root of both sides. Since distances are nonnegative, AQ = BQ.
Help students understand the Converse of the
Perpendicular Bisector Theorem by having them
make a poster showing the theorem and its converse.
Then have them explain what is given as the
hypothesis of the theorem and of its converse, and
what they have to prove for the theorem and its
converse.
Your Turn
7.
_
_
_
AD is 10 inches long. BD is 6 inches long. Find the length of AC.
D
Since D is equidistant from A and C and ¯
DB is perpendicular
AC by the diagram, then ¯
DB must be the perpendicular
to ¯
AC and AC = 2 · BC. BC 2 + 6 2 = 10 2, so BC = 8 in.
bisector of ¯
and AC = 16 in.
C
A
B
Proving Theorems about Right Angles
Explain 3
‹ › _
−
The symbol ⊥ means that two figures are perpendicular. For example, ℓ ⊥ m or XY ⊥ AB.
Example 3

If two lines intersect to form one right angle, then they are perpendicular
and they intersect to form four right angles.
Given: m∠1 = 90°
Proving Theorems About Right
Angles
1 2
4 3
Prove: m∠2 = 90°, m∠3 = 90°, m∠4 = 90°
Statement
QUESTIONING STRATEGIES
Reason
1. m∠1 = 90°
1. Given
2. ∠1 and ∠2 are a linear pair.
If a linear pair of angles has equal measure,
why are the angles right angles? Since a linear
pair of angles is supplementary, their sum must be
180°. So each angle must be half of 180°, or 90°.
2. Given
3. ∠1 and ∠2 are supplementary.
3. Linear Pair Theorem
4. m∠1 + m∠2 = 180°
4. Definition of supplementary angles
5. 90° + m∠2 = 180°
5. Substitution Property of Equality
6. m∠2 = 90°
6. Subtraction Property of Equality
7. m∠2 = m∠4
© Houghton Mifflin Harcourt Publishing Company
7. Vertical Angles Theorem
8. m∠4 = 90°
8. Substitution Property of Equality
9. m∠1 = m∠3
9. Vertical Angles Theorem
10. m∠3 = 90°

EXPLAIN 3
Prove each theorem about right angles.
10. Substitution Property of Equality
If two intersecting lines form a linear pair of angles with equal measures, then the lines are
perpendicular.
Given: m∠1 = m∠2
By the diagram, ∠1 and ∠2 form a linear pair so ∠1 and ∠2 are supplementary
by the Linear Pair Theorem . By the definition of supplementary angles,
ℓ
1
Prove: ℓ ⊥ m
2
m
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Technology
MP.5 Encourage students to use the geometry
software to create linear pairs of angles and then
measure them. Then have them construct
perpendicular lines and measure each of the angles to
verify that they are right angles.
m∠1 = m∠2
m∠1 + m∠2 = 180° . It is also given that
,
so m∠1 + m∠1 = 180° by the Substitution Property of Equality . Adding
gives 2 ⋅ m∠1 = 180°, and m∠1 = 90° by the Division Property of Equality. Therefore,
∠1 is a right angle and ℓ ⊥ m by the definition of perpendicular lines .
Module 19
968
Lesson 4
DIFFERENTIATE INSTRUCTION
IN1_MNLESE389762_U8M19L4 968
Visual Cues
9/23/14 9:09 AM
Use colors to identify the steps in an indirect proof. Write each of the steps in a
different color. Then place colored boxes around the parts of the proof that
correspond to each step. For example, write the step “Assume the opposite of the
conclusion is true” in blue and put a blue box around the assumption made in the
indirect proof.
Perpendicular Lines 968
Reflect
ELABORATE
8.
QUESTIONING STRATEGIES
State the converse of the theorem in Part B. Is the converse true?
If two intersecting lines are perpendicular, then they form a linear pair of angles with
equal measures; yes.
How is constructing a perpendicular bisector
related to constructing a segment
bisector? The construction of a perpendicular
bisector starts with a segment.
Your Turn
9.
Given: b ǁ d, c ǁ e , m∠1 = 50°, and m∠5 = 90°. Use the diagram to find m∠4.
a
How is constructing a perpendicular bisector
related to constructing a perpendicular to a
point on a line? The construction of a perpendicular
to a point on a line starts with marking equal
distances from the point along the line. This creates
a segment with the point as the midpoint of the
segment.
c
4
d
2
3
5
6
m∠4 = 40°; by corresponding angles because c ǁ e, m∠1 = m∠2, and
by vertical angles, m∠2 = m∠3, so m∠3 = 50°; because m∠5 = 90°,
then a ⊥ d and m∠3 + m∠4 = 90, so m∠4 = 40°.
e
Elaborate
10. Discussion Explain how the converse of the Perpendicular Bisector Theorem justifies
the compass-and-straightedge construction of the perpendicular bisector of a segment.
SUMMARIZE THE LESSON
C
A
B
D
© Houghton Mifflin Harcourt Publishing Company
If you are given a line and a point P, how do you
construct a line that is perpendicular to the given line
using a compass and straightedge? Sample answer:
Use a compass to locate two points A and B on the
line that are the same distance from point P. Then
_
construct the perpendicular bisector of AB.
1
b
The construction involves making two arcs that intersect in two points. Each of these two
intersection points is equidistant from the endpoints of the segment, because the arcs are
the same radius. So, both of the intersection points are on the perpendicular bisector of
the segment.
11. Essential Question Check-In How can you construct perpendicular lines and
prove theorems about perpendicular bisectors?
Constructing a line perpendicular to a given line involves using a compass to locate two
points that are not on the given line but are equidistant from two points on the given line.
You can prove the Perpendicular Bisector Theorem using a reflection and its properties,
and you can prove the Converse of the Perpendicular Bisector Theorem using an indirect
argument involving the Pythagorean Theorem.
Module 19
969
Lesson 4
LANGUAGE SUPPORT
IN1_MNLESE389762_U8M19L4 969
Visual Cues
Have students work in pairs. Give students pictures of intersecting, parallel, skew,
and perpendicular lines. Instruct one student in each pair to explain why a pair of
lines is or is not perpendicular, and prove it to the partner. Have the student who
is not explaining write notes about the proof explanation. Then have students
switch roles.
969
Lesson 19.4
9/23/14 9:16 AM
EVALUATE
Evaluate: Homework and Practice
1.
• Online Homework
• Hints and Help
• Extra Practice
P
How can you construct a line perpendicular to
line ℓ that passes through point P using paper
folding?
ℓ
Fold line ℓ onto itself so that the crease passes through point P.
The crease is the required perpendicular line.
2.
4.
Check for Reasonableness How can you use 3.
a ruler and a protractor to check the construction
in Elaborate Exercise 11?
‹ ›
−
Use the ruler to check that CD bisects ¯
AB.
‹ ›
−
¯
Use the protractor to check that AB and CD
Describe the point on the perpendicular bisector
of a segment that is closest to the endpoints of the
segment.
are perpendicular.
closest to the endpoints of the segment.
The midpoint of the segment is the point
on the perpendicular bisector that is
Represent Real-World Problems A field of soybeans is
watered
by a rotating irrigation system. The watering arm,
_
CD, rotates around its center point. To show the area of the
crop of soybeans that will be watered, construct a circle with
diameter CD.
D
_
Use_
the_
diagram to find the lengths. BP is the
_ perpendicular bisector
of AC. CQ is the perpendicular bisector of BD. AB = BC = CD.
Q
P
A
_
Suppose AP = 5 cm. What is the length of PC?
6.
B
C
D
Suppose AP
_= 5 cm and BQ = 8 cm. What is the
length of QD?
By the Perpendicular Bisector Theorem,
By the Perpendicular Bisector Theorem,
PC = 5 cm.
QD = 8 cm.
Module 19
Exercise
IN1_MNLESE389762_U8M19L4 970
Depth of Knowledge (D.O.K.)
Mathematical Practices
MP.5 Using Tools
2–4
2 Skills/Concepts
MP.5 Using Tools
5–13
2 Skills/Concepts
MP.6 Precision
3 Strategic Thinking
MP.6 Precision
2 Skills/Concepts
MP.6 Precision
17
3 Strategic Thinking
MP.2 Reasoning
18
3 Strategic Thinking
MP.3 Logic
19
3 Strategic Thinking
MP.2 Reasoning
14
15–16
Concepts and Skills
Practice
Explore
Constructing Perpendicular
Bisectors and Perpendicular Lines
Exercises 1–4
Example 1
Proving the Perpendicular Bisector
Theorem
Exercises 5–8,
15–16
Example 2
Proving the Converse of the
Perpendicular Bisector Theorem
Exercises 9–11,
14
Example 3
Proving Theorems about Right
Angles
Exercises 12–13,
15–16
AVOID COMMON ERRORS
A common error when working with perpendicular
lines is to assume that lines are perpendicular if they
look perpendicular. Emphasize the need to establish
that there are right angles before saying that lines are
perpendicular.
Lesson 4
970
1 Recall of Information
1
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©sima/
Shutterstock
C
5.
ASSIGNMENT GUIDE
4/19/14 12:40 PM
Perpendicular Lines 970
7.
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Critical Thinking
MP.3 Ask students to compare constructions using
a reflective device, tracing paper, compass and
straightedge, and geometry software. Discuss the
advantages of each. Have students consider not only
how easy each is to use, but also how accurate, and
how well each tool helps them understand the
geometric relationships being studied.
Suppose AC =_
12 cm and QD = 10 cm. What is
the length of QC?
8.
Suppose PB =_3 cm and AD = 12 cm . What is
the length of PC?
1⋅
QC = 8 cm; AC = 12 cm so BC = _
2
AC = 6 cm and CD = 6 cm. By the
By the Pythagorean Theorem, PC 2 = PB 2 + BC 2,
Pythagorean Theorem, QD 2 = QC 2 + CD 2,
so PC 2 = 3 2 + 4 2 and PC = 5 cm.
PC = 5 cm; AD = 12 cm, so AB = BC = CD = 4 cm.
so 10 2 = QC 2 + 6 2 and QC = 8 cm.
Given: PA = PC and BA = BC. Use the diagram to find the lengths or angle measures
described.
9.
Suppose m∠2 = 38°. Find m∠1.
‹ ›
−
m∠1 = 52°; because PA = PC and BA = BC, then PB is
the perpendicular bisector of ¯
AC, and m∠PBC = 90°.
Then m∠1 = 90° - 38° = 52°.
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Technology
MP.5 Ask each student to draw a simple sketch that
10. Suppose PA = 10 cm and
_ PB = 6 cm.
What is the length of AC?
A
3
1 2
4 B
C
11. Find m∠3 + m∠4.
AC = 16 cm; because PA = PC and BA = BC,
‹ ›
−
then PB is the perpendicular bisector of ¯
AC and
involves perpendicular lines and right angles. Have
students exchange sketches and attempt to reproduce
the one they receive using geometry software.
P
Theorem, PA 2 = PB 2 + BA 2, so 10 2 = 6 2 + BA 2
m∠3 + m∠4 = 90°; because P and B are
‹ ›
−
equidistant from the endpoints, then PB
_
is the perpendicular bisector of AC.
‹ ›
‹ › −
−
So, PB ⊥ AC and the lines meet at right
and BA = 8 cm. Then AC = 2 ⋅ BA = 16 cm.
angles, and m∠3 + m∠4 = 90°.
△PBA is a right triangle. By the Pythagorean
Given: m ǁ n, x ǁ y, and y ⊥ m. Use the diagram to find the angle measures.
© Houghton Mifflin Harcourt Publishing Company
m
n
2 a
1 3
6 4
5
x
7
y
8
12. Suppose m∠7 = 30°. Find m∠3.
y ⊥ m, so m∠7 + m∠8 = 90° and m∠8 = 60°.
13. Suppose m∠1 = 90°.
What is m∠2 + m∠3 + m∠5 + m∠6?
Because m∠1 = 90°, then x ⊥ n and the
Using alternate interior angles, because x ǁ y,
lines intersect at four right angles. So,
then m∠8 = m∠6, so m∠6 = 60°. ∠6 and ∠3
m∠2 + m∠3 + m∠5 + m∠6 = 180°.
are vertical angles, so m∠3 = 60°.
Module 19
IN1_MNLESE389762_U8M19L4 971
971
Lesson 19.4
971
Lesson 4
9/23/14 9:17 AM
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Math Connections
MP.1 Remind students about how to use the
Use this diagram of trusses for a railroad bridge in Exercise 14.
D
A
3 4
E
F
1 2
B
C
Perpendicular Bisector Theorem and its converse to
write equations that will help them find angle
measures related to perpendicular lines.
_
_
14. Suppose BE is the perpendicular bisector of DF. Which of the following statements do you
know are true? Select all that apply. Explain your reasoning.
A. BD = BF
B. m∠1 + m∠2 = 90°
_
C. E is the midpoint of DF. A, C, and D; the given information that ¯
BE is the perpendicular bisector
D. m∠3 + m∠4 = 90°
_ _
E. DA ⊥ AC
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Modeling
MP.4 Have students explain how they can use a
of ¯
DF means that both points E and B are equidistant from D and F so
answer choices A and C are true. Then, ¯
EB ⊥ ¯
DF and m∠3 + m∠4 = 90°,
so answer choice D is known to be true. The given information does not tell anything about
¯
DA or ¯
AC, though, so answer choices B and E may or may not be true.
protractor or reflective device to check the accuracy
of their constructions.
15. Algebra Two lines intersect to form a linear pair with equal measures. One angle has the
measure 2x° and the other angle has the measure (20y - 10)°. Find the values of x and y.
Explain your reasoning.
x = 45, y = 5; the linear pair formed has equal measures, so the lines are
perpendicular and then 2x° = 90°, so x = 45, and (20y - 10)° = 90°, so y = 5.
16. Algebra Two lines intersect to form a linear pair of congruent angles. The measure
15y
of one angle is (8x + 10)° and the measure of the other angle is ___ °. Find the values
( )
2
of x and y. Explain your reasoning.
© Houghton Mifflin Harcourt Publishing Company
x = 10, y = 12; the linear pair formed are congruent angles, so the lines are
perpendicular and then (8x + 10)° = 90°, so x = 10, and
15y °
= 90°, so y = 12.
(___
2 )
H.O.T. Focus on Higher Order Thinking
17. Communicate Mathematical Ideas The valve pistons
on a trumpet are all perpendicular to the lead pipe. Explain
why the valve pistons must be parallel to each other.
The valve pistons are lines that are perpendicular
to the same line (the lead pipe), so they form right
angles with the same line. By the converse of the
corresponding angles theorem, all the congruent
right angles mean the valve pistons are parallel to
each other.
Module 19
IN1_MNLESE389762_U8M19L4 972
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lead pipe
valve pistons
Lesson 4
09/06/15 12:04 PM
Perpendicular Lines 972
18. Justify Reasoning Prove the theorem: In a plane, if a transversal
is perpendicular to one of two parallel lines, then it is perpendicular
to the other.
_ _
_ _
_ _
Prove: RS ⊥ AB
Given: RS ⊥ CD and AB ∥ CD
COLLABORATIVE LEARNING
ACTIVITY
Have students work in groups of three or four. Ask
them to choose one student to give directions.
Instruct the other students to each draw a line and a
point that is not on the line. The first student will
then give a set of directions, step by step, for
constructing a perpendicular to a line through a
point that is either on the line or not on the line. The
other students will not know which construction it is
until they have followed the directions. Once the
student has successfully guided the other students
through the construction process, have another
student take the lead and describe the construction
to the others.
Statements
_
CD
1. AB ǁ ¯
1. Given
2. m∠RTD = m∠RVB
2. Corresponding Angles Theorem
RS ⊥ ¯
CD
3. ¯
3. Given
4. m∠RTD = 90°
4. Definition of perpendicular lines
5. m∠RVB = 90°
5. Substitution Property of Equality
RS ⊥ ¯
AB
6. ¯
T
D
A
V
S
B
6. Definition of perpendicular lines
Given: ∠1 and ∠2 are supplementary.
Prove: ∠1 and ∠2 cannot both be obtuse.
Assume that two supplementary angles can both be obtuse angles. So, assume that
∠1 and ∠2
are both obtuse . Then m∠1 > 90° and m∠2 > 90°
the definition of obtuse angles . Adding the two inequalities,
by
m∠1 + m∠2 > 180° . However, by the definition of supplementary angles,
Have students draw and mark a set of figures to
illustrate the Perpendicular Bisector Theorem and its
converse. Have students provide explanatory captions
for the figures.
m∠1 + m∠2 = 180° . So m∠1 + m∠2 > 180° contradicts the given
information.
This means the assumption is false , and therefore
© Houghton Mifflin Harcourt Publishing Company
∠1 and ∠2 cannot both be obtuse .
Module 19
IN1_MNLESE389762_U8M19L4 973
Lesson 19.4
C
19. Analyze Mathematical Relationships Complete the indirect proof to show that
two supplementary angles cannot both be obtuse angles.
JOURNAL
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Reasons
R
973
Lesson 4
9/23/14 9:26 AM
Lesson Performance Task
AVOID COMMON ERRORS
A utility company wants to build a wind farm to provide electricity to the towns of Acton,
Baxter, and Coleville. Because of concerns about noise from the turbines, the residents of all
three towns do not want the wind farm built close to where they live. The company comes to
an agreement with the residents to build the wind farm at a location that is equally distant from
all three towns.
Students can use proportions to find the actual
distances between towns, but they may set them up
incorrectly. To find the distance between Acton and
Coleville, they can write and solve this proportion:
1 in. = ______
1.5 in. . The correct pattern is
_____
10 mi
x mi
map distance
map distance
_____________
= _____________.
actual distance actual distance
Acton
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Critical Thinking
MP.3 Marcus said that it isn’t necessary to draw
1.5 in.
120°
Coleville
4 in.
Scale 1 in. : 10 mi
Baxter
three perpendicular bisectors to find the location of
the wind farm. Two is sufficient, he said. Was he
right? Explain. Yes; sample answer: Any two of the
perpendicular bisectors will intersect at a point.
That point must be the location of the wind farm
because the two lines can intersect in only one
point. The third perpendicular bisector provides a
useful check on the accuracy of the constructions of
the first two perpendicular bisectors.
a. Use the drawing to draw a diagram of the locations of the towns using a scale of
1 in. : 10 mi. Draw the 4-inch and 1.5-inch lines with a 120° angle between them.
Write the actual distances between the towns on your diagram.
b. Estimate where you think the wind farm will be located.
c. Use what you have learned in this lesson to find the exact location of the wind
farm. What is the approximate distance from the wind farm to each of the
three towns?
© Houghton Mifflin Harcourt Publishing Company
a. Distances: Acton–Baxter, about 49 miles; Baxter–Coleville, 40 miles;
Acton–Coleville, 15 miles
b. Student answers will vary.
c. Students should construct perpendicular bisectors of the three
lines of the triangle. The point of intersection of the bisectors is the
point that is equidistant from the three vertices of the triangle. Any
two of the bisectors is sufficient to locate the point, but a third one
is useful for spotting possible errors in the drawing of the first
two bisectors.
Module 19
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Lesson 4
EXTENSION ACTIVITY
IN1_MNLESE389762_U8M19L4 974
Pose this challenge to students: Three other towns have signed up to obtain
electricity from the wind farm. All are the same distance from the farm as Acton,
Baxter, and Coleville. On your drawing, show three points that could be the
locations of the towns. Explain how you found the points and how you know they
meet the conditions of the problem.
Students should draw a circle with its center at the wind farm and passing
through Acton, Baxter, and Coleville. They can choose any three points on the
circle as the locations of the three new towns. Those points must be the same
distance from the farm as Acton, Baxter, and Coleville because all radii of a circle
are equal in length.
4/19/14 12:40 PM
Scoring Rubric
2 points: Student correctly solves the problem and explains his/her reasoning.
1 point: Student shows good understanding of the problem but does not fully
solve or explain his/her reasoning.
0 points: Student does not demonstrate understanding of the problem.
Perpendicular Lines 974